The Keller-Segel system with logistic growth and signal-dependent motility

Abstract

The paper is concerned with the following chemotaxis system with nonlinear motility functions \begin{equation}\label{0-1}\tag{$\ast$} \begin{cases} u_t=\nabla \cdot (γ(v)\nabla u- uχ(v)\nabla v)+μu(1-u), &x\in Ω, ~~t>0, 0=Δv+ u-v,& x\in Ω, ~~t>0,\\ u(x,0)=u_0(x), & x\in Ω, \end{cases} \end{equation} with homogeneous Neumann boundary conditions in a bounded domain $\Omega \subset {\mathbb{R}}^2$ with smooth boundary, where the motility functions $\gamma (v)$ and $\chi (v)$ satisfy the following conditions \begin{itemize} \item {\color{black}$(\gamma ,\chi )\in [C^2[0,\infty ){]}^2$} with $\gamma (v)>0$ and {\color{black} $\frac{|\chi (v){|}^2}{\gamma (v)}$ is bounded for all $v\ge 0$.} %for all $v\ge 0$ and $\underset{v\to \infty }{\lim }\frac{|\chi (v){|}^2}{\gamma (v)}$ exists. \end{itemize} By employing the method of energy estimates , we establish the existence of globally bounded solutions of \eqref{0-1} with $\mu >0$ for any $u_0\in W^{1,\infty }(\Omega )$. Then based on a Lyapunov function, we show that all solutions $(u,v)$ of \eqref{0-1} will exponentially converge to the unique constant steady state $(1,1)$ provided $\mu >\frac{K_0}{16}$ with $K_0=\underset{0\le v\le \infty }{\max }\frac{|\chi (v){|}^2}{\gamma (v)}$.

References in corpus (1)

• Comparison Methods for a Keller--Segel-type Model of Pattern Formations with Density-suppressed Motilities